William M. Briggs

Statistician to the Stars!

There Is No Such Thing As An Average Man

Several readers—thanks to James, Steve E, and Ted Poppke—found the story “When U.S. air force discovered the flaw of averages“, which is an excerpt from the Todd Ross book The End of Average.

Story is this. Mid-last-century, an Air Force—ooh rah!—lieutenant was dispatched to measure body shapes of pilots. Everything “including thumb length, crotch height, and the distance from a pilot’s eye to his ear” was put to the tape, and, as statistical practice goes, averages were taken.

The lieutenant (Daniels) was experienced in his duty and had measured college students before.

Even more surprising, when Daniels averaged all his data, the average hand did not resemble any individual’s measurements. There was no such thing as an average hand size…

Using the size data he had gathered from 4,063 pilots, Daniels calculated the average of the 10 physical dimensions believed to be most relevant for design, including height, chest circumference and sleeve length. These formed the dimensions of the “average pilot,” which Daniels generously defined as someone whose measurements were within the middle 30 per cent of the range of values for each dimension. So, for example, even though the precise average height from the data was five foot nine, he defined the height of the “average pilot” as ranging from five-seven to five-11.

Lo, none of the four thousand pilots fit the average on all the dimensions.

Daniels discovered that if you picked out just three of the ten dimensions of size — say, neck circumference, thigh circumference and wrist circumference — less than 3.5 per cent of pilots would be average sized on all three dimensions. Daniels’s findings were clear and incontrovertible. There was no such thing as an average pilot. If you’ve designed a cockpit to fit the average pilot, you’ve actually designed it to fit no one.

There’s more to the story about a supposed ideal female that you’ll want to scan.

Why isn’t there an average pilot? Suppose height was the sole engineering concern. Then 30 percent of the men measured would be average, because the middle 30 percent would be the middle 30 percent, n’est-ce pas?

Next suppose right arm length and height were of concern. Now if whatever caused a man’s height also, and in step (speaking loosely), caused his arm length, then every man’s height would track his arm length, and again 30 percent of the men measured would be average.

But since there are many causes of height (genetics plus environment), and many different, plus some similar, causes of arm length, then the two measures won’t track exactly. Because the causes differ, and although there will be some overlap, the same 30 percent of men in the middle of height won’t be the precise set of men in the 30 percent middle of arm length.

As the number of dimensions increases, the causes become more diverse and the men more unlike one another (across all the dimensions). Simple as that.

It would be a tremendous but common mistake to speak of “normal distributions”, “correlations”, and so forth to say why men don’t match the mean. None of these statistical abstractions are real. The measurements are real, the causes of the things measured are real. But “normal distributions” and “correlations” are not real.

Thus it is a mistake to say “height and arm length are normally distributed” and another mistake to say “height and arm length have such-and-such a correlation.” Nothing in the world is “normally distributed” because normal distributions don’t exist.

Our uncertainty in the unknown heights and arm lengths of future pilots (and not the ones already measured) might, at crude approximation only, be represented using a normal distribution, and the uncertainty we have in the relationship between these two measures might crudely be summarized using a correlation, but that’s all statistics can do. It remains mute on what the causes of the measurements are.

Now no one man may fit the fuzzy average across several dimensions, but it could still be that there may be groups who may who cluster around other measures beside the mean. That is, no man might be within plus-or-minus 30 percent of the average-of-all-dimensions, but some men might be within some plus-or-minus 30 percent of some-function-of-all-dimensions (which isn’t the average). What might these functions be?

Well, anything. Could be height multiplied by arm length divided by inter-eye distance all added to elbow thickness is one function which identifies a large number of similar men, who we could then say fit a type.

Since the number of possible functions increases with the number of dimensions measured, it becomes quite a chore to find representative ones. There are many functions in math! So if we want to do this algorithmically, we have to limit the functions to certain kinds (addition and multiplication, say). Or if we want to do a real bangup job, we could hunt for the various causes of the dimensions.

15 Comments

  1. Briggs, are you suggesting exploratory factor analysis?

  2. Briggs

    March 24, 2016 at 9:49 am

    Gary,

    Gaack! No. I’m suggesting trying to understand what it is that makes bodies of different types, of finding functional relationships between dimensions that make biological and physical sense. Factor analysis is nothing but repeated linear combinations of the dimensions. It may well chance upon some relationships, but it’s far, far too overused.

  3. Briggs,
    Agreed that FA can’t identify cause. But it does boil down a multitude of dimensions and can be useful as long as it isn’t taken to far. Many years ago I used it to identify associations in species assemblages and the results were informing. Of course, wisdom in the use of any tool must reside with the craftsman, not the tool.

  4. When I first read the story it made me think of the urban legend of the space pen where the story goes that NASA spent millions of dollars to develop a pen that could write in zero gravity. A young wag asked the NASA spokesman what the Soviets did. The spokesman replied, “They used a pencil.”

    It’s difficult to conceive the stubbornness required to force fit a single seat to a non-existent average man when an adjustable seat was required. It’s instructive that average is still thought of and used this way today across many disciplines.

  5. Hmmm, Chi-square distribution anyone? (That doesn’t exist either, but its useful for coming up with a predictive distribution) Also: http://www.johndcook.com/blog/2011/09/01/multivariate-normal-shell/

  6. The Procrustean solution.

    There is much that has to be rediscovered. They should have consulted a tailor or a blacksmith. Suits of medieval armour must have been fitted to the knight.

  7. From an aerospace perspective shaving weight off is so critical that good ideas like adjustable seats will have a hard time being considered. It’s much easier to yell at a new hire to figure out the one sites fits all uncomfortably approach.

    Never mind an adjustable seat that must survive ejection!

    Bill R,

    Neat!

  8. To be fair we must note that your link refers to a time shortly after the Second World War when the volume and speed of production of aircraft precluded such luxuries as a custom fit.

    https://m.youtube.com/watch?v=Ht_i1EiZHyc

  9. If I recall correctly, you had to be between 5’6″ and 6’6″ to be a fighter pilot in the Navy. If you were shorter you couldn’t reach the controls and if you were taller you wouldn’t fit into the cockpit. This was a problem when the Navy decided to let women become fighter pilots, most were too short. The fighters don’t have adjustable controls or seats. The DoD is now correcting the problem by designing combat gear specifically for women. This is going to be a challenge. For example, the Army has body armor that will stop a .30 caliber armor piercing bullet, but it is heavy, over 30 lbs. Boron carbide plates aren’t light. I am wondering how they will make lighter armor for the women that will still stop the AP bullets.

  10. The real story being the “Flaw of Averages” article is that the USAF tried to find an average sized person…and because of that we’ve got car seats that adjust. Some vehicles now have pedals that adjust.

    If a govt bureaucracy could behave rationally with such data and get it implemented so quickly, there’s still hope. Though I wonder if the USAF would, if confronted today with a comparable analytical issue, still behave so rationally. That is, do such bureaucracies behave like entropy–only getting “larger”/worse?

    On the other hand, the multi-variate complexity of non-global-warming models suggest that these too are making a similar fundamental error in condensing things down to a single average number for the whole planet (as opposed to regional, seasonal, day/night, etc. averages). A comprehensive analysis would be interesting from this perspective to, again, rebut the alarmist assertions…though, no doubt, that would have no effect on the alarmist’s faith their warming/climate-change/disruption religion.

  11. The air force could issue customised clip-in seat inserts to pilots, as part of their uniform.

  12. Ken, it’s too bad that pilots had to die in order for a government bureaucracy to behave rationally. It’s also questionable as to whether it was the loss of life or the loss of military assets that motivated the bureaucracy.

    I agree that it’s questionable how the USAF would respond today, although I tend to think it would be more prone to look for the space pen solution instead of the pencil.

    Some times duct tape is still the best solution. 😉
    http://www.imdb.com/video/screenplay/vi763560217

  13. In the olden days when statistical calculations were done with the Monroe mechanical monster, one of the urban (academic?) legends was that Mother Nature was lognormal. Did the USAF try log-transformed measurements?

  14. I wonder how much money the military deemed necessary to discover the obvious.

    JMJ

  15. Scotian,
    “Suits of medieval armour must have been fitted to the knight.”
    It was.
    and helms are planished using tree stumps.
    Body armour that didn’t fit it was adjusted and remade until it was comfortable.
    an under-pading garments jacket acted as the memory foam of the day.
    The coat hardy was pinned from only four pieces of fabric to the person and the seems stitched afterwards so that’s how the clothing looked beautifully fitted or sprayed on at times.

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